With Visa and MasterCard, no. They both have rules that limit stores from requiring you to show your ID as a condition of purpose. These rules also make them accept your card even if you refuse to show your ID. However, you may still be required to show ID for other reasons related to the sale, such as for buying alcohol/tobacco, using a card flagged as lost/stolen, suspected fraud, etc.

If someone insists that you show your ID to use your credit card, you can show a copy of the Visa or MasterCard rules.

Often nearly impossible to see, a skimmer is a small device placed in or over a card reader (ATM, gas pump, checkout) to record information from your credit or debit card's magnetic stripe.  That information can then be used to make fraudulent purchases and sometimes even a clone of your card.